Begin Notes Immediately. Look at Example Below!!! Glue in Notebook

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1 Begin Notes Immediately Look at Eample Below!!! Glue in Notebook

2 Graphing Rational Functions

3 The Parent Function can be transformed by using f( ) 1 f ( ) a k h What do a, h and k represent? a the vertical stretch or compression h the horizontal translation k the vertical translation

4 What is an asymptote? The line where a graph approaches There can be vertical asymptotes horizontal asymptotes and What is the vertical asymptote? What is the horizontal asymptote? y 0 slant asymptotes.

5 1 g ( ) 3 Eample 1 What is the transformation? Down 3 units What is the vertical asymptote? 0 What is the horizontal asymptote? y 3

6 1 g ( ) 3 Eample 1 What is the Domain? 0,0 0, What is the Range? y 3, 3 3,

7 Eample g ( ) 1 What is the transformation? Left units What is the vertical asymptote? What is the horizontal asymptote? y 0

8 To Graph Rational Functions 1. Factor Completely. Vertical Asymptotes: set the denominator equal to zero. It is stated as an equation # The vertical asymptotes are the domain eclusions. 3. Horizontal Asymptotes: compare the degrees of the numerator and denominator.

9 Horizontal Asymptote If low deg ree high deg ree 3 5 then HA: y = 0 If same same deg ree deg ree you look at y leading coefficient leading coefficient then HA: y = If high deg ree low deg ree then there is NO horizontal asymptote HA: none

10 Graphing continued 4. Roots: set the numerator = 0 What is another name for the -intercepts? Roots What does it mean on the graph? Where the graph crosses the -ais

11 5. Holes: Graphing continued Look at factors that are common to numerator and denominator. Set those equal to 0 and solve. Vertical Asymptotes do NOT include the holes.

12 Eample 3 6 Identify the following for Vertical asymptote Horizontal Asymptote Roots: Holes: 1 none 3, none Domain:, 1 1, Range: Y-intercept: 1, 0, 6 3/18/015 1:14 PM 8-4: Graphs of Rational Functions [Day 1] 1

13 Identify the following for Vertical asymptote 0 Horizontal Asymptote look at the degrees Eample 4 Roots 0 Same Degree Divide the leading coefficients. Vertical asymptote Horizontal Asymptote Roots: Holes: y 1 none Y-intercept: 0, 1 y 1 3/18/015 1:14 PM 8-4: Graphs of Rational Functions [Day 1] 13

14 Eample 4 graph = Zeros: Vertical Asymptote: VA: Horizontal Asymptote: HA: y 1 y = 1 Domain:,, Range:,11, 8-4: Graphs of Rational Functions [Day 1]

15 Eample 5 Identify the following for Vertical asymptote none Horizontal Asymptote look at the degrees f ( ) 1 none Roots 10 1 Holes 10 1 Vertical asymptote Horizontal Asymptote Roots: Holes: none none 1 Y-intercept: 1 1, 0,1 3/18/015 1:14 PM 8-4: Graphs of Rational Functions [Day 1] 15

Eample 5 1 1 zeros : 1, VA: None, HA: None, Hole at 1 (1,) y int ercept : (0,1)

16 Eample zeros : 1, VA: None, HA: None, Hole at 1 (1,) y int ercept : (0,1) Domain :,1 1, 1 Range :,, y 3/18/015 1:14 PM Graphs of Rational Functions [Day ] 16

17 Eample 6 Identify the following for Vertical asymptote 0 Horizontal Asymptote look at the degrees 1 y 1 Roots 0 Holes 10 1 Vertical asymptote Horizontal Asymptote Roots: Holes: 0 y 1 1 (1, 1) 3/18/015 1:14 PM 8-4: Graphs of Rational Functions [Day 1] 17

18 Eample 6 3 Hole at 1 (1, 1) zeros :, VA: 0, HA: y 1, Domain : (-,0) (0,1) (1, ) All Reals ecept 0, 1 Range : (-,-1) (-1,1) (1, ) All Reals ecept -1, 1 3/18/015 1:14 PM Graphs of Rational Functions [Day ] 18

19 Etra problems following this slide

20 Eample 7 Given, 3graph, determine the zeros, all asymptotes/holes, and domain zeros :, VA:, HA : y 1, Hole at 1 Domain : All Reals ecept 1, 3/18/015 1:14 PM Graphs of Rational Functions [Day ] 0

21 Eample 8 9 Given, 3 graph, determine the zeros and all asymptotes/holes zeros : 3, VA: None, HA: None, Hole at 3 Domain : All Reals ecept 3 Hole at = 3 3/18/015 1:14 PM Graphs of Rational Functions [Day ] 1

22 Without a Calculator Identify the zeros and determine all of the asymptotes to the following: 1. = = ( + )( 3) = + 3/18/015 1:14 PM Graphs of Rational Functions [Day ]

23 Eample 3 Identify the zeros and vertical asymptote(s) of 34 3 Zeros : 4, 1 Vertical Asymptote : 3 3/18/015 1:14 PM 8-4: Graphs of Rational Functions [Day 1] 3

24 Horizontal Asymptotes: compare the degrees of the numerator and denominator. Degree of Numerator > Degree of Denominator Top bigger---no means NO horizontal asymptotes. Must find Oblique asymptote. Degree of Denominator > Degree of Numerator Bottom bigger---oh means horizontal asymptote is y=0. Degree of Numerator = Degree of Denominator -CO means find ratio of leading coefficients lead coefficient of numerator y lead coefficient of denom

25 Identify the following for Eample 5 ( 9) 4 ( 3) 3 Vertical asymptote 0 0 Horizontal Asymptote look at the degrees ( 9) 1 4 y Roots ( 9) 0 ( 3)( 3) 0 3 Same Degree Divide the leading coefficients. Vertical asymptote Horizontal Asymptote Roots: Holes: y 3 none 3/18/015 1:14 PM 8-4: Graphs of Rational Functions [Day 1] 5

26 Eample 5 Given ( 9) 4 = = y = Zeros: 3 Vertical Asymptote: VA: Horizontal Asymptote: HA : y Domain: All Real Numbers ecept and Range: y and y5 3/18/015 1:14 PM 8-4: Graphs of Rational Functions [Day 1] 6

Domain: The domain of f is all real numbers except those values for which Q(x) =0.

Domain: The domain of f is all real numbers except those values for which Q(x) =0. Math 1330 Section.3.3: Rational Functions Definition: A rational function is a function that can be written in the form P() f(), where f and g are polynomials. Q() The domain of the rational function such